[[1]}] A tree decomposition of a graph \(G=(V,E)\)  is a pair \(\langle T,\chi \rangle \) , where \(T=(N,F)\)  is a tree, and \(\chi \)  is a labelling function
assigning with each vertex \(p\in N\)  a set of vertices \(\chi (p)\subseteq V\)  such that the following conditions are satisfied: (1) for each node
\(b\)  of \(G\) , there exists \(p\in N\)  such that \(b\in \chi (p)\) ; (2) for each edge \((b,d)\in E\) , there exists \(p\in N\)  such that \(\lbrace b,d\rbrace \subseteq \chi (p)\) ; and, (3) for each node \(b\)  of \(G\) , the set \(\lbrace p\in N \mid b\in \chi (p)\rbrace \)  induces a connected subtree of \(T\)  (connectedness
condition).
The width of \(\langle T,\chi \rangle \)  is the number \(\max _{p\in N}(|\chi (p)|-1)\) . The treewidth of \(G\) , denoted by \(tw(G)\) , is the minimum
width over all its tree decompositions. \(\Box \) 

